Find all the turning points of y=x^{1/x} for x>0 and decide
whether each is a maximum or minimum. Give a sketch of the graph.

Quick Route

Stage: 5 Challenge Level:

The solution below was sent in by Taryn from Kerang Technical High
School. The quickest route takes $14$ minutes walking exactly one
quarter of the way along the edge and then in a straight line
across the field. Congratulations to all of you who found this
solution. Other good solutions were sent in by Jenny from KJS, by
Thomas and by Andrei from Tudor Vianu National College, Romania.

As $x\leq 1$ the critical time is given by $x = 0.25 \;
\text{km} = 250 \; \text{m}$. This gives a time of $14$ minutes to
cross the field. Increasing and decreasing $x$ slightly increases
the time taken so this is a minimum time.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.